Rational Generalized Nash Equilibrium Problems

TL;DR

This paper introduces a novel hierarchy of rational optimization problems and applies Moment-SOS relaxations to compute or detect the nonexistence of generalized Nash equilibria in non-convex rational function settings.

Contribution

It develops a new approach using rational expressions and Moment-SOS relaxations to analyze and compute GNE in non-convex rational problems, extending existing methods.

Findings

Hierarchy can compute GNE if it exists

Method can detect nonexistence of GNE

Numerical experiments demonstrate efficiency

Abstract

This paper studies generalized Nash equilibrium problems that are given by rational functions. The optimization problems are not assumed to be convex. Rational expressions for Lagrange multipliers and feasible extensions of KKT points are introduced to compute a generalized Nash equilibrium (GNE). We give a hierarchy of rational optimization problems to solve rational generalized Nash equilibrium problems. The existence and computation of feasible extensions are studied. The Moment-SOS relaxations are applied to solve the rational optimization problems. Under some general assumptions, we show that the proposed hierarchy can compute a GNE if it exists or detect its nonexistence. Numerical experiments are given to show the efficiency of the proposed method.